Turek Cylinder Benchmark

updated 17 November 2006 (old version here)

Testcase 3D-Z1 from Schaefer, Turek. See there for details about geometry etc.

The solution is steady, and it is found by running the time dependent solvers with 1st order BDF (=backward Euler) until t=10, about where a stationary state is reached. The timestep is chosen as equal to the largest mesh width, which is 12.5 times the smallest mesh width. With a maximal inflow velocity of 0.45, this would give a maximal CFL of 5.625. Note that the velocity is low where the mesh is fine (i. e. near the cylinder).

The drag and the lift have been evaluated by numerical integration of the stress on the cylinder. The pressure difference was calculated using a L² scalar product with two regularized delta functions located at the points of interest. The mass error is the integral over time of |inflow-outflow|. Implementation details can be found in life-playground/benchmark/cylinder/turek. I didn't want to put the finest mesh into cvs, you can find it here.

See the current results here, with the bounds indicated by Schaefer and Turek:

Solver	Ndof	h=Δt	drag	lift	Δp	mass error	cpu sec >per timestep<th>	Memory / MB
lower bound			6.05	0.0080	0.165
upper bound			6.25	0.0100	0.175
PC P1bubble-P1	36540	0.2	6.19	-0.0043	0.189	2.0e-6	42.9	46
	165236	0.1	6.10	-0.0094	0.172	2.3e-6	432.0	174
	693449	0.05	6.14	0.0231	0.176	1.9e-6	3376.2	697
(stopped at t=6.625)	3121877	0.025	6.18	0.0063	0.171	n.a.	46909.7	3144
PC P2-P1	45675	0.2	6.18	-0.0052	0.169	1.3e-6	204.1	200
	203796	0.1	6.17	0.0097	0.169	2.7e-5	4253.1	859
	853694	0.05	6.18	0.0090	0.171	4.4e-4	29704.8	1853
IP P1-P1	8340	0.2	6.54	0.0200	0.177	3.9e-6	3.5	90
	36444	0.1	6.25	-0.0039	0.171	6.0e-5	15.5	389
	152216	0.05	6.27	0.0205	0.176	5.7e-4	62.6	3224
	672080	0.025	6.25	0.0111	0.173	3.4e-3	281.8	3705
IP P1-P1 γβ=0	8340	0.2	6.30	0.0337	0.176	3.3e-6	1.7
	36444	0.1	6.16	-0.0082	0.171	5.5e-5	7.0	389
	152216	0.05	6.23	0.0221	0.175	5.8e-4	25.3	1648
	672080	0.025	6.24	0.0116	0.173	3.4e-3	110.3	3692
IP P2-P2	58120	0.2	6.24	0.0174	0.174	2.2e-4	47.0	1358
SD P1-P1	8340	0.2	6.81	-0.0753	0.178	2.7e-4	2.1	52
	36444	0.1	6.35	0.0122	0.170	8.1e-4	12.5	205
	152216	0.05	6.23	0.0203	0.171	1.9e-3	65.0	996
	672080	0.025	6.21	0.0093	0.170	6.3e-3	369.8	5821

Meshes used

h	hcyl	vertices	cells	edges
0.2	0.016	2085	9400	12445
0.1	0.008	9111	42930	55784
0.05	0.004	38054	180411	233826
0.025	0.002	168020	816599

Observations

• Higher order elements were tested on linear meshes, so the geometry will induce dominating errors (not iso-parametric).
• PC P1 Bubble-P1: The drag and pressure difference values are within the bounds for coarse meshes already. The lift coefficient seems well approximated on the finest mesh only, being too small though.
• PC P2-P1: Has drag and pressure within bounds from the coarsest mesh on. The lift is within bounds if the mesh is not too coarse.
• IP P1-P1: Drag and pressure difference are within the bounds or very close for all but the coarsest meshes. The lift seems well approximated on the finest mesh only, being too large though.
• IP P1-P1 without stabilization of the convection: It is not really needed at Re=20. Setting gamma_beta=0 gives better values for the drag (now within bounds from the second mesh) and is faster by a factor 2!
• IP P2-P2: Gives correct values for drag and pressure difference already on the coarsest mesh. Given the general difficulty of approximating the lift on coarse meshes (see also Schaefer, Turek), nothing can be said about the lift. Running a finer mesh is not possible (memory!)
• SD P1-P1: Drag and lift overestimated on coarse meshes, but all values are within the bounds for the finest mesh!
• CPU time IP and SD: With respect to degrees of freedom, IP and SD scale only slightly superlinear in time.
• CPU time PC: Although accelerated by a factor of about 4 using bi-CGSTAB for the innermost system (not rerun for P2-P1), computation times for PC are still much bigger and scale less nicely with the number of degrees of freedom.
• Mass error: Error remains constant under mesh refinement for PC, whereas IP and SD have increasing error under mesh refinement. This may be linked to the coupled solving and addressed by decreasing the solver tolerance. Note that decreasing the solver tolerance is not costly as the system is well preconditioned.
• Memory cost: Comparable (for the same mesh) for all solvers.